Remarks about a generalized pseudo-relativistic Hartree equation

TL;DR

This paper proves the existence of ground state solutions for a generalized pseudo-relativistic Hartree equation with nonlinearities, and discusses the regularity of solutions under certain hypotheses.

Contribution

It establishes existence and regularity results for solutions to a generalized pseudo-relativistic Hartree equation with specific nonlinear conditions.

Findings

Existence of ground state solutions under certain hypotheses.

Regularity properties of solutions.

Applicability to a range of nonlinearities.

Abstract

With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[(-\Delta+m^2)^\sigma u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $0<\sigma<1$, $V$ is a bounded continuous potential and $F$ the primitive of $f$. We also show results about the regularity of any solution of this problem.